# iid and correlated order statistics a comparison

Consider the two random variables $X_1$ and $X_2$ defined via i.i.d, non-negative random variables $Y_1$ and $Y_2$ as

$X_1=\min(Y_1,Y_2)\\$

$X_2=\min(Y_1,Y_1)$ – (maximally correlated)

The question is can we say $X_1 \leq X_2$ always? How to prove it?

#### Solutions Collecting From Web of "iid and correlated order statistics a comparison"

If you have written the question correctly, then $X_2=Y_1$ (minimum of two functions is the function itself. Remember that random variables are functions).

Then it’s always the case that $X_1 \leq X_2$ as just by definition, minimum of two functions is always less than or equal to each of the two functions. Alternatively, $\min(Y_1,Y_2)=\dfrac{Y_1+Y_2-|Y_1-Y_2|}{2}$, and show that this expression is always less than or equal to $Y_1$